I'm not sure if this will be considered on-topic, let's see what happens. I've asked this previously in Physics SE and it was closed as needing focus. It is now past 30 days and too late to consider migration.

note that I'm asking about how to treat the atoms and phonons in the wall and how these affect the scattering, not about simulating the subsequent free motion in the box.

For example Singh et al. (2009) J. App. Phys. 106, 024314 Modeling of subcontinuum thermal transport across semiconductor-gasinterfaces seems to address this using a Boltzman distribution for the gas atoms in the vacuum and for phonons in the wall material rather than simulating individual atoms of the walls. However I can’t see how to apply that microscopically on a per-collision basis without estimating how many phonons are involved; is it one phonon on average? More? Less?

Figures at the end of The Interaction of Molecular Beams with Solid Surfaces show that the spread in angles of atoms incident at 45 degrees does get wider with increased surface temperature, but there is no information on changes in kinetic energy, but Wikipedia's Helium atom scattering; Inelastic measurements suggests that interactions with phonons can lead to a much more complicated picture.

Background: I'm writing a simple Monte-Carlo simulation for an ultra-high vacuum chamber. I'll initially distribute atoms randomly in position and direction and with speeds derived from a Maxwell-Boltzmann distribution. At one end there is a tube of variable length; once a particle passes the far end of it it will be considered "pumped" and no longer tracked.

The idea is to address the effect of a long narrow tube between the chamber and the pump on the effective pumping speed.

I'll assume a monatomic gas with no internal degrees of freedom.

Question: I can introduce the effect of wall roughness by randomly varying the microscopic surface normal from the macroscopic normal at each collision, but I can't think of a way to handle the random momentum transfer effects due to thermal vibrations of the atoms in the wall. These can both change the kinetic energy and the direction of scattering.

What would be a simple, first principles way to introduce it?

I can imagine treating the atom in the wall as a free particle with a similar thermal distribution of speed and direction, but the atoms of the wall are not free, they're constrained by bonds to adjacent atoms.

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    $\begingroup$ I think that questions can't be migrated to private beta sites. This would make sense since not all private beta sites last the full 3 weeks, if the activity level is not high enough. If we can recruit a little more and encourage more questions/day, we should be fine though. $\endgroup$ – Nike Dattani May 13 '20 at 4:36
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    $\begingroup$ Are you treating the gas molecules quantum mechanically or classically? $\endgroup$ – taciteloquence May 27 '20 at 8:29
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    $\begingroup$ @taciteloquence classically for sure. I'm just looking for a way to model the variation in the recoil velocity and direction from elastic scattering from a flat surface that has some basis in physics, and apart from roughness, I think that phonons in the surface are the most important source of this. $\endgroup$ – uhoh May 27 '20 at 8:56
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    $\begingroup$ @taciteloquence I never know how to respond to "you shouldn't want to know the answer to your question". It may turn out that adding physics to the scattering process doesn't affect the results too much, but I would still like to know how to do it so I can try it and see what difference it makes. I'm not a "maybe it won't matter, so I'll leave it out" person. When atoms scatter from surfaces they exchange energy and momentum with the surface, let's find out how to model it! $\endgroup$ – uhoh May 27 '20 at 10:30
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    $\begingroup$ I agree! Totally didn't mean to imply that your question was illegitimate. I have at least a kernel of an answer I can post below. $\endgroup$ – taciteloquence May 27 '20 at 10:33

Whatever scattering mechanism you choose must respect detailed balance in equilibrium: on average, the number of particles hitting a patch of the wall at a given angle and velocity must equal the number of particles reflected at the same angle and velocity. If this were not the case, the system would not be in equilibrium.

There are numerous ways to do this including

  • Have each particle scatter specularly.
  • Have each particle that hits the wall keep its energy but choose a random direction from a cosine distribution. (Depending on the context, this is known as Lambert's cosine law or the Knudsen cosine law.) This is diffuse scattering.
  • Randomize the energy as well by drawing a new energy from the Maxwell-Boltzmann distribution.*

These methods have different meanings. The first means that the surface is perfectly smooth and doesn't transfer energy. The second means that the surface is so rough that all knowledge about the particles previous direction is lost. The third means the surface thermalizes the particle, so knowledge about the previous energy is also lost.

There are many variations on this. For example, a particle could scatter specularly with some probability p and scatter diffusely with probability 1-p. This indicates that the surface is somewhere between perfectly smooth and super rough.

Now, your system is not in equilibrium, but you still should choose a method that respects detailed balance in equilibrium. I'm guessing that the inside of the vacuum chamber is relatively smooth. I'm also guessing that the walls of the chamber won't do much to thermalize the chamber on each collision. (Think about a small gas cylinder with cold air in it. The atoms hit the walls a gazillion times a second, but it still takes a while for the air to warm up. So, individual collisions don't cause much thermalization.)

You could probably do something like reflect specularly with a large probability, reflect diffusely with a smaller probability, and thermalize with a very small probability. (You could do a back-of-the envelope calculation for how many collisions it takes to thermalize a gas particle on average.)

Now, none of these are first-principles methods, but I don't think that first-principles methods are meaningful unless you know the properties of the chamber walls in great detail. It sounds like that's not the case, so I'd use a simpler model and be done with it. I'm sure that simulations used for real-world engineering generally don't use first principle approaches.

* This is surprisingly easy to do wrong. The methods I mentioned are all also used to model electromagnetic radiation. In particular, the third method is basically blackbody radiation with gas molecules instead of light. So, you can get a good reference on how to do this with light (e.g. Radiative Heat Transfer by Michael Modest) and just substitute a Maxwell-Boltzmann distribution for gas particles wherever you see a Bose-Einstein distribution for photons. I'm sure there are good books on gas flows too, but I'm not familiar with them.

  • $\begingroup$ +10. Nice first answer, and we hope to see much more of you !!! $\endgroup$ – Nike Dattani May 29 '20 at 23:49

Since no one has responded with expertise, I'll attempt a speculative answer here.

To my mind, the simplest model of the atoms on the surface of the walls would be an ensemble of independent classical 3D harmonic oscillators at some temperatures $T_{\rm wall}$. That would be pretty easy to describe from a numerical standpoint, since their velocity could be described by the Boltzmann distribution. Then you could treat the collisions with the wall with classical energy/momentum conservation. This would still allow transfer of momentum and energy between the walls and gas.

Caveat: Treating the wall as independent harmonic oscillators like this will mean that your simulation does not conserve energy, since the walls will acts as a thermal bath. In practice, this is probably a good thing since the temperature of the walls will be set by the room temperature or any heating you do.

  • $\begingroup$ Okay great! So in this case my atoms "bounce" off of wall atoms which have a distribution of velocities corresponding to a classical harmonic oscillator, that's nice. I think all I need now is a guess for the potential constant $k$. $\endgroup$ – uhoh May 27 '20 at 11:39
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    $\begingroup$ You could probably put some bounds on $k$ by using the mass of Fe (or whatever your walls are made of) and making sure that the amplitude of the oscillations is small relative to the atomic spacing of the wall material. (At least that would be a place to start, not guarantees it is helpful). $\endgroup$ – taciteloquence May 27 '20 at 11:48
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    $\begingroup$ I added a caveat to my answer: under this approximation, energy will not be conserved. $\endgroup$ – taciteloquence May 28 '20 at 2:22

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